A013972 -id:A013972 - cp's OEIS frontend

A211347 Numbers n such that n = sigma_k(m) for some k >= 1.

1, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 17, 18, 20, 21, 24, 26, 28, 30, 31, 32, 33, 36, 38, 39, 40, 42, 44, 48, 50, 54, 56, 57, 60, 62, 63, 65, 68, 72, 73, 74, 78, 80, 82, 84, 85, 90, 91, 93, 96, 98, 102, 104, 108, 110, 112, 114, 120, 121, 122

Offset: 1

Jon Perry, Feb 05 2013

nonn

Sigma_k(n) = Sum[d|n, d^k].
Sigma_0(n) can be any positive integer and so is ignored in this sequence.
The asymptotic density of this sequence is 0 (Niven, 1951, Rao and Murty, 1979). - Amiram Eldar, Jul 23 2020

			Sigma_2(4) = 1 + 4 + 16 = 21 so 21 is in the sequence.

Giovanni Resta, Table of n, a(n) for n = 1..10000
Ivan Niven, The asymptotic density of sequences, Bull. Amer. Math. Soc., Vol. 57 (1951), pp. 420-434.
R. Sita Rama Chandra Rao and G. Sri Rama Chandra Murty, On a theorem of Niven, Canadian Mathematical Bulletin, Vol 22, No. 1 (1979), pp. 113-115.
Eric W. Weisstein, MathWorld: Divisor function

Cf. A000203, A001157, A001158, A001159, A001160.
Cf. A013954, A013955, A013956, A013957, A013958, A013959, A013960, A013961, A013962, A013963, A013964, A013965, A013966, A013967, A013968, A013969, A013970, A013971, A013972.
Cf. A000005.

upto[n_] := Select[Union@Flatten[{1, DivisorSigma[Range@Max[1,Floor@Log[#,n]], #] & /@ Range[2,n]}], # <= n &]; upto[122] (* Giovanni Resta, Feb 05 2013 *)

list(lim)=if(lim<3, return(if(lim<1,[],[1]))); my(v=List([1])); for(k=1,logint((lim\=1)-1,2), forfactored(m=2,sqrtnint(lim-1,k), my(t=sigma(m,k)); if(t<=lim, listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Apr 09 2022

A013964 a(n) = sigma_16(n), the sum of the 16th powers of the divisors of n.

Original entry on oeis.org

1, 65537, 43046722, 4295032833, 152587890626, 2821153019714, 33232930569602, 281479271743489, 1853020231898563, 10000152587956162, 45949729863572162, 184887084343023426, 665416609183179842, 2177986570740006274, 6568408508343827972, 18447025552981295105

Offset: 1

N. J. A. Sloane

If the canonical factorization of n into prime powers is the product of p^e(p) then sigma_k(n) = Product_p ((p^((e(p)+1)*k))-1)/(p^k-1).

Sum_{d|n} 1/d^k is equal to sigma_k(n)/n^k. So sequences A017665-A017712 also give the numerators and denominators of sigma_k(n)/n^k for k = 1..24. The power sums sigma_k(n) are in sequences A000203 (k=1), A001157-A001160 (k=2,3,4,5), A013954-A013972 for k = 6,7,...,24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for sequences related to sigma(n).

Cf. A000203, A001157-A001160, A013675, A013954-A013972, A017665-A017712.

[DivisorSigma(16, n): n in [1..20]]; // Vincenzo Librandi, Sep 10 2016

DivisorSigma[16, Range[30]] (* Vincenzo Librandi, Sep 10 2016 *)

my(N=99, q='q+O('q^N)); Vec(sum(n=1, N, n^16*q^n/(1-q^n))) \\ Altug Alkan, Sep 10 2016

a(n) = sigma(n, 16); \\ Amiram Eldar, Oct 29 2023

[sigma(n,16)for n in range(1,14)] # Zerinvary Lajos, Jun 04 2009

G.f.: Sum_{k>=1} k^16*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003
Dirichlet g.f.: zeta(s-16)*zeta(s). - Ilya Gutkovskiy, Sep 10 2016
From Amiram Eldar, Oct 29 2023: (Start)
Multiplicative with a(p^e) = (p^(16*e+16)-1)/(p^16-1).
Sum_{k=1..n} a(k) = zeta(17) * n^17 / 17 + O(n^18). (End)

A013968 a(n) = sigma_20(n), the sum of the 20th powers of the divisors of n.

Original entry on oeis.org

1, 1048577, 3486784402, 1099512676353, 95367431640626, 3656161927895954, 79792266297612002, 1152922604119523329, 12157665462543713203, 100000095367432689202, 672749994932560009202, 3833763649708914645906, 19004963774880799438802, 83668335217551100221154

Offset: 1

N. J. A. Sloane

If the canonical factorization of n into prime powers is the product of p^e(p) then sigma_k(n) = Product_p ((p^((e(p)+1)*k))-1)/(p^k-1).

Sum_{d|n} 1/d^k is equal to sigma_k(n)/n^k. So sequences A017665-A017712 also give the numerators and denominators of sigma_k(n)/n^k for k = 1..24. The power sums sigma_k(n) are in sequences A000203 (k=1), A001157-A001160 (k=2,3,4,5), A013954-A013972 for k = 6,7,...,24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sigma(n).

Cf. A000203, A001157-A001160, A013954-A013972, A017665-A017712.

[DivisorSigma(20,n): n in [1..50]]; // G. C. Greubel, Nov 03 2018

DivisorSigma[20,Range[20]] (* Harvey P. Dale, Jul 26 2015 *)

vector(50, n, sigma(n,20)) \\ G. C. Greubel, Nov 03 2018

[sigma(n,20)for n in range(1,13)] # Zerinvary Lajos, Jun 04 2009

G.f.: Sum_{k>=1} k^20*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003
From Amiram Eldar, Oct 29 2023: (Start)
Multiplicative with a(p^e) = (p^(20*e+20)-1)/(p^20-1).
Dirichlet g.f.: zeta(s)*zeta(s-20).
Sum_{k=1..n} a(k) = zeta(21) * n^21 / 21 + O(n^22). (End)

A013970 a(n) = sigma_22(n), the sum of the 22nd powers of the divisors of n.

Original entry on oeis.org

1, 4194305, 31381059610, 17592190238721, 2384185791015626, 131621735227521050, 3909821048582988050, 73786993887028445185, 984770902214992292491, 10000002384185795209930, 81402749386839761113322, 552061570551763831158810, 3211838877954855105157370

Offset: 1

N. J. A. Sloane

If the canonical factorization of n into prime powers is the product of p^e(p) then sigma_k(n) = Product_p ((p^((e(p)+1)*k))-1)/(p^k-1).

Sum_{d|n} 1/d^k is equal to sigma_k(n)/n^k. So sequences A017665-A017712 also give the numerators and denominators of sigma_k(n)/n^k for k = 1..24. The power sums sigma_k(n) are in sequences A000203 (k=1), A001157-A001160 (k=2,3,4,5), A013954-A013972 for k = 6,7,...,24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sigma(n).

Cf. A000203, A001157-A001160, A013954-A013972, A017665-A017712.

[DivisorSigma(22,n): n in [1..50]]; // G. C. Greubel, Nov 03 2018

lst={};Do[AppendTo[lst,DivisorSigma[22,n]],{n,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Mar 11 2009 *)
a[ n_] := DivisorSigma[ 22, n]; (* Michael Somos, Dec 19 2016 *)

vector(50, n, sigma(n,22)) \\ G. C. Greubel, Nov 03 2018

[sigma(n,22)for n in range(1,12)] # Zerinvary Lajos, Jun 04 2009

G.f.: Sum_{k>=1} k^22*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003
From Amiram Eldar, Oct 29 2023: (Start)
Multiplicative with a(p^e) = (p^(22*e+22)-1)/(p^22-1).
Dirichlet g.f.: zeta(s)*zeta(s-22).
Sum_{k=1..n} a(k) = zeta(23) * n^23 / 23 + O(n^24). (End)

A017669 Numerator of sum of -3rd powers of divisors of n.

Original entry on oeis.org

1, 9, 28, 73, 126, 7, 344, 585, 757, 567, 1332, 511, 2198, 387, 392, 4681, 4914, 757, 6860, 4599, 1376, 2997, 12168, 455, 15751, 9891, 20440, 3139, 24390, 147, 29792, 37449, 4144, 22113, 6192, 55261, 50654, 15435, 61544, 7371, 68922, 172, 79508, 24309, 10598

Offset: 1

N. J. A. Sloane

Sum_{d|n} 1/d^k is equal to sigma_k(n)/n^k. So sequences A017665-A017712 also give the numerators and denominators of sigma_k(n)/n^k for k = 1..24. The power sums sigma_k(n) are in sequences A000203 (k=1), A001157-A001160 (k=2,3,4,5), A013954-A013972 for k = 6,7,...,24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001

			1, 9/8, 28/27, 73/64, 126/125, 7/6, 344/343, 585/512, 757/729, 567/500, 1332/1331, 511/432, ...

G. C. Greubel, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Vincenzo Librandi)

Cf. A017670 (denominator), A002117, A013662.

[Numerator(DivisorSigma(3,n)/n^3): n in [1..40]]; // G. C. Greubel, Nov 08 2018

Table[Numerator[DivisorSigma[-3, n]], {n, 50}] (* Vladimir Joseph Stephan Orlovsky, Jul 21 2011 *)
Table[Numerator[DivisorSigma[3, n]/n^3], {n, 1, 40}] (* G. C. Greubel, Nov 08 2018 *)

vector(40, n, numerator(sigma(n, 3)/n^3)) \\ G. C. Greubel, Nov 08 2018

Numerators of coefficients in expansion of Sum_{k>=1} x^k/(k^3*(1 - x^k)). - Ilya Gutkovskiy, May 24 2018
From Amiram Eldar, Apr 02 2024: (Start)
sup_{n>=1} a(n)/A017670(n) = zeta(3) (A002117).
Dirichlet g.f. of a(n)/A017670(n): zeta(s)*zeta(s+3).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017670(k) = zeta(4) (A013662). (End)

A017670 Denominator of sum of -3rd powers of divisors of n.

Original entry on oeis.org

1, 8, 27, 64, 125, 6, 343, 512, 729, 500, 1331, 432, 2197, 343, 375, 4096, 4913, 648, 6859, 4000, 1323, 2662, 12167, 384, 15625, 8788, 19683, 2744, 24389, 125, 29791, 32768, 3993, 19652, 6125, 46656, 50653, 13718, 59319, 6400, 68921, 147, 79507, 21296, 10125

Offset: 1

N. J. A. Sloane

Sum_{d|n} 1/d^k is equal to sigma_k(n)/n^k. So sequences A017665-A017712 also give the numerators and denominators of sigma_k(n)/n^k for k = 1..24. The power sums sigma_k(n) are in sequences A000203 (k=1), A001157-A001160 (k=2,3,4,5), A013954-A013972 for k = 6,7,...,24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001

			1, 9/8, 28/27, 73/64, 126/125, 7/6, 344/343, 585/512, 757/729, 567/500, 1332/1331, 511/432, ...

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A017669.

[Denominator(DivisorSigma(3,n)/n^3): n in [1..40]]; // G. C. Greubel, Nov 08 2018

Table[Denominator[DivisorSigma[-3, n]], {n, 50}] (* Vladimir Joseph Stephan Orlovsky, Jul 21 2011 *)
Table[Denominator[DivisorSigma[3, n]/n^3], {n, 1, 40}] (* G. C. Greubel, Nov 08 2018 *)

vector(40, n, denominator(sigma(n, 3)/n^3)) \\ G. C. Greubel, Nov 08 2018

Denominator of Sum_{d|n} 1/d^3.

Denominators of coefficients in expansion of Sum_{k>=1} x^k/(k^3*(1 - x^k)). - Ilya Gutkovskiy, May 24 2018

A017671 Numerator of sum of -4th powers of divisors of n.

Original entry on oeis.org

1, 17, 82, 273, 626, 697, 2402, 4369, 6643, 5321, 14642, 3731, 28562, 20417, 51332, 69905, 83522, 112931, 130322, 85449, 196964, 124457, 279842, 179129, 391251, 242777, 538084, 46839, 707282, 218161, 923522, 1118481, 1200644, 41761, 1503652, 604513, 1874162

Offset: 1

N. J. A. Sloane

Sum_{d|n} 1/d^k is equal to sigma_k(n)/n^k. So sequences A017665-A017712 also give the numerators and denominators of sigma_k(n)/n^k for k = 1..24. The power sums sigma_k(n) are in sequences A000203 (k=1), A001157-A001160 (k=2,3,4,5), A013954-A013972 for k = 6,7,...,24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001

			1, 17/16, 82/81, 273/256, 626/625, 697/648, 2402/2401, 4369/4096, 6643/6561, 5321/5000, ...

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A017672 (denominator), A013662, A013663.

[Numerator(DivisorSigma(4,n)/n^4): n in [1..40]]; // G. C. Greubel, Nov 08 2018

Table[Numerator[DivisorSigma[-4, n]], {n, 50}] (* Vladimir Joseph Stephan Orlovsky, Jul 21 2011 *)
Table[Numerator[DivisorSigma[4, n]/n^4], {n, 1, 40}] (* G. C. Greubel, Nov 08 2018 *)

vector(40, n, numerator(sigma(n, 4)/n^4)) \\ G. C. Greubel, Nov 08 2018

Numerators of coefficients in expansion of Sum_{k>=1} x^k/(k^4*(1 - x^k)). - Ilya Gutkovskiy, May 24 2018
From Amiram Eldar, Apr 02 2024: (Start)
sup_{n>=1} a(n)/A017672(n) = zeta(4) (A013662).
Dirichlet g.f. of a(n)/A017672(n): zeta(s)*zeta(s+4).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017672(k) = zeta(5) (A013663). (End)

A017672 Denominator of sum of -4th powers of divisors of n.

Original entry on oeis.org

1, 16, 81, 256, 625, 648, 2401, 4096, 6561, 5000, 14641, 3456, 28561, 19208, 50625, 65536, 83521, 104976, 130321, 80000, 194481, 117128, 279841, 165888, 390625, 228488, 531441, 43904, 707281, 202500, 923521, 1048576, 1185921, 39304, 1500625, 559872, 1874161

Offset: 1

N. J. A. Sloane

Sum_{d|n} 1/d^k is equal to sigma_k(n)/n^k. So sequences A017665-A017712 also give the numerators and denominators of sigma_k(n)/n^k for k = 1..24. The power sums sigma_k(n) are in sequences A000203 (k=1), A001157-A001160 (k=2,3,4,5), A013954-A013972 for k = 6,7,...,24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001

			1, 17/16, 82/81, 273/256, 626/625, 697/648, 2402/2401, 4369/4096, 6643/6561, 5321/5000, ...

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A017671.

[Denominator(DivisorSigma(4,n)/n^4): n in [1..40]]; // G. C. Greubel, Nov 08 2018

Table[Denominator[DivisorSigma[-4, n]], {n, 50}] (* Vladimir Joseph Stephan Orlovsky, Jul 21 2011 *)
Table[Denominator[DivisorSigma[4, n]/n^4], {n, 1, 40}] (* G. C. Greubel, Nov 08 2018 *)

vector(40, n, denominator(sigma(n, 4)/n^4)) \\ G. C. Greubel, Nov 08 2018

Denominators of coefficients in expansion of Sum_{k>=1} x^k/(k^4*(1 - x^k)). - Ilya Gutkovskiy, May 24 2018

A017673 Numerator of sum of -5th powers of divisors of n.

Original entry on oeis.org

1, 33, 244, 1057, 3126, 671, 16808, 33825, 59293, 51579, 161052, 64477, 371294, 69333, 254248, 1082401, 1419858, 652223, 2476100, 1652091, 4101152, 120789, 6436344, 687775, 9768751, 6126351, 14408200, 317251, 20511150, 349591, 28629152, 34636833, 13098896

Offset: 1

N. J. A. Sloane

Sum_{d|n} 1/d^k is equal to sigma_k(n)/n^k. So sequences A017665-A017712 also give the numerators and denominators of sigma_k(n)/n^k for k = 1..24. The power sums sigma_k(n) are in sequences A000203 (k=1), A001157-A001160 (k=2,3,4,5), A013954-A013972 for k = 6,7,...,24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001

			1, 33/32, 244/243, 1057/1024, 3126/3125, 671/648, 16808/16807, 33825/32768, 59293/59049, ...

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A017674 (denominator), A013663, A013664.

[Numerator(DivisorSigma(5,n)/n^5): n in [1..40]]; // G. C. Greubel, Nov 08 2018

Table[Numerator[DivisorSigma[-5, n]], {n,100}] (* Vladimir Joseph Stephan Orlovsky, Jul 22 2011 *)
Table[Numerator[DivisorSigma[5, n]/n^5], {n, 1, 40}] (* G. C. Greubel, Nov 08 2018 *)

vector(40, n, numerator(sigma(n, 5)/n^5)) \\ G. C. Greubel, Nov 08 2018

Numerators of coefficients in expansion of Sum_{k>=1} x^k/(k^5*(1 - x^k)). - Ilya Gutkovskiy, May 25 2018
From Amiram Eldar, Apr 02 2024: (Start)
sup_{n>=1} a(n)/A017674(n) = zeta(5) (A013663).
Dirichlet g.f. of a(n)/A017674(n): zeta(s)*zeta(s+5).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017674(k) = zeta(6) (A013664). (End)

A017674 Denominator of sum of -5th powers of divisors of n.

Original entry on oeis.org

1, 32, 243, 1024, 3125, 648, 16807, 32768, 59049, 50000, 161051, 62208, 371293, 67228, 253125, 1048576, 1419857, 629856, 2476099, 1600000, 4084101, 117128, 6436343, 663552, 9765625, 5940688, 14348907, 307328, 20511149, 337500, 28629151, 33554432, 13045131

Offset: 1

N. J. A. Sloane

Sum_{d|n} 1/d^k is equal to sigma_k(n)/n^k. So sequences A017665-A017712 also give the numerators and denominators of sigma_k(n)/n^k for k = 1..24. The power sums sigma_k(n) are in sequences A000203 (k=1), A001157-A001160 (k=2,3,4,5), A013954-A013972 for k = 6,7,...,24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001

			1, 33/32, 244/243, 1057/1024, 3126/3125, 671/648, 16808/16807, 33825/32768, 59293/59049, ...

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A017673.

[Denominator(DivisorSigma(5,n)/n^5): n in [1..40]]; // G. C. Greubel, Nov 08 2018

Table[Denominator[DivisorSigma[-5, n]], {n,100}] (* Vladimir Joseph Stephan Orlovsky, Jul 22 2011 *)
Table[Denominator[DivisorSigma[5, n]/n^5], {n, 1, 40}] (* G. C. Greubel, Nov 08 2018 *)

vector(40, n, denominator(sigma(n, 5)/n^5)) \\ G. C. Greubel, Nov 08 2018

Denominators of coefficients in expansion of Sum_{k>=1} x^k/(k^5*(1 - x^k)). - Ilya Gutkovskiy, May 25 2018

cp's OEIS Frontend

A211347 Numbers n such that n = sigma_k(m) for some k >= 1.

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A013964 a(n) = sigma_16(n), the sum of the 16th powers of the divisors of n.

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A013968 a(n) = sigma_20(n), the sum of the 20th powers of the divisors of n.

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A013970 a(n) = sigma_22(n), the sum of the 22nd powers of the divisors of n.

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A017669 Numerator of sum of -3rd powers of divisors of n.

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A017670 Denominator of sum of -3rd powers of divisors of n.

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A017671 Numerator of sum of -4th powers of divisors of n.

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